grade12Maths JIT Document 2017

JUST IN TIME MATERIAL GRADE 12 TERM 1 - 2017 KZN DEPARTMENT OF EDUCATION CURRICULUM GRADES 10 – 12 DIRECTORATE TERM  1   TOPIC   PAGE   CALCULUS   2  -­‐  29   ANALYTICAL  GEOMETRY   30  -­‐  47   TRIGONOMETRY   48  -­‐  110   REFERENCES   110   1

Topic CALCULUS Weighting Sub-topics/Clarification 1.   An intuitive understanding of the limit concept. Related Different symbolic forms of representing the derivative of a function “Gradient at a point”, eg , f’(x) =…... concepts/terms/vocabula ry Prior-knowledge/ •   Sketching of parabola and straight line. Background knowledge •   Gradient formula (average gradient) •   Substitution of values •   Factorisation Resources Calculator Educator Activity Demonstrate the sketching of parabola and straight line functions on the same set of axes having two points of intersection. Learner Activities Sketching of parabola and straight line functions on the same set of axes having two points of intersection Methodology •   Use a function as example, e.g. 𝑓 𝑥 = 3𝑥 + 7. Find e.g. 𝑓(1), 𝑓(5), 𝑓(−2), 𝑓(𝑡), 𝑓(𝑥 + 1), 𝑓(𝑥 + ℎ). •   Then give learners a short practice exercise. See Annexure A . Learners may do 3 from numbers 1 to 4 of this Annexure. 2

•   Draw a sketch graph of any curve 𝑦 = 𝑓 𝑥 . •   Consider two points on the curve: 𝑥0 ; 𝑦0 and 𝑥2 ; 𝑦2 . The average gradient between these two points is the same as the gradi 4 64 through the two points, namely: 5 7 . 85 687 45 647 9 8:; 69 8 •   Show learners that = , 85 687 ; if the distance between 𝑥0 and 𝑥2 is h (  i. e. let  𝑥0 = 𝑥 and  𝑥2 = 𝑥 + ℎ). •   Using the sketch graph show learners how the gradient of the secant gets closer and closer to the gradient of the tangent to the cu smaller and smaller. •   Now the concept of limit is needed: If h becomes very, very small the secant will almost almost fall on the tangent and the gradient of the secant will be equal to the 9 8:; 69 8 say that in the limit when h tends to 0 (ℎ → 0), will be the gradient of the tangent to the curve at that point. ; •   h cannot just be made equal to zero because the fraction will then be undefined, AND because for calculating gradient you need •   So: the gradient of the tangent to a curve at a certain point is lim f (x + h) − f (x) . h→0 h f (x + h) − f (x ) •   lim is also called the derivative of the function 𝑓 𝑥 and is given the symbol 𝑓 C 𝑥 . h→0 h 3

ANNEXURE A 1. Given: ACO = 3 - 2x. Determine: a (3) [2] b A(O) [2] ¢ Aad) [2] d AC-3) [2] e h(a) [2] f h(a- 2) [3] g h(3-a) [3] h h(x+a) [3] 2. Given: f (x) = 2x’ + x - 6. Determine: a f(-3) [2] b f(-1) [2] c f(1) [2] d f(2) [2] e f(m) [2] Ff f(n+2) [3] g fl-a) [3] h f+h) [3] i f(-2)-f@3)_ [4] j f(a—3)+fla+3) [4] k fix+h)-fo) — [3] 3 Given: g(x) = . Determine: a g(2) [2] b g(-16) [2] c eG) [2] d a(-F {2] e g(a) fi} f g(a-2) fi] g g(a+1) 11] oh g(x+a) [1] i g(-8)+9(16) [3] j glm+2)+g(m) [4] k gix+h)-g@) [4] 4 Given: k(x) = 2x’. Determine: a k(-3) (2] b k(-1) [2] ce k(o) [2] d k(2) [2] e k@ {1] f k(a+2) [3] g ka-1) [3] h k(x +h) (3] i k@2)+k) ~~ =~B] j kK(a+2)+k(a) [3] k k(x+h)-k(x) [3]

Week Day 2 Topic CALCULUS Weighting ±5 marks Sub-topics/Clarification F Rules for differentiation Related concepts/terms/vocabulary Limit ,gradient, substitution Prior-knowledge/ Calculation of gradient; factorisation; simplification. Background knowledge Resources textbook Educator Activity •   In an intuitive way, by referring back to the first principle examples “deduce” the rule: D 𝑎𝑥 F = 𝑎𝑛𝑥 F60 . D8 •   Work through some examples with learners of how to determine the derivative using rules. Learners then need to practice more of the same. See Annexure B below. Learners activity 1.   f ( x) = − x 2 2.   f ( x) = 3x 2 + 5x 3.   𝑓 𝑥 = 2𝑎𝑥 2 − 3𝑥 0 4.   𝑓 𝑥 = 𝑥 2   2 5.   𝑓 𝑥 = 8 3 6.   f(x) = 5x 7.   𝑔 𝑥 = 6 Assessment Homework from textbook. 5

Errors/Misconceptions/ Wrong notation i.e f(x) = instead of f’ (x)= Problem Areas Wrong substitution e.g f(x) = lim f (x+h) – f(x ) h→0 h f(x) = lim (x+h)2 – (x2 +1 ) h→0 h and incorrect simplification. h≠0 but h→0 Beefing Up Non routine application of f’(x) = lim f (x+h) – f(x ) h→0 h 6

Annexure B RULES FOR DIFFERENTIATON Complete the following 𝑓 𝑥 = 𝑥F 𝑓C 𝑥 = 𝑓 𝑥 = 𝑚𝑥 F 𝑓C 𝑥 = 𝑓 𝑥 = 𝑘, where k is a constant 𝑓C 𝑥 = Class Activity D4 1. Determine D8 1.1. 𝑦 = 3𝑥 + 4 1.2. 𝑦 = 𝑥(𝑥 − 2) 1.3. 𝑦 = 𝑥 M 1.4. 𝑦 = − N 8 O8 N 628 5 :P8:00 1.5. 𝑦 = 8 1.6. 𝑥𝑦 + 𝑦 =   𝑥 2 − 𝑥 − 2 2. Find 𝑓 C (𝑥) 7

8 N 68 5 628 2.1. 𝑓 𝑥 = 862 0 2 2.2. 𝑓 𝑥 =   𝑥 − 8 N 2.3. 𝑓 𝑥 = 8𝑥 02 P 2.4. 𝑓 𝑥 = − N 5 8 862M 2.5. 𝑓 𝑥 = 86M 2P4 N 6R 2.6 𝑓 𝑦 = 26S4 D4 2.7 Consider 𝑦 = (𝑥 + 1)2 show that =2 𝑦 D8 4 2.8 If 𝑦 = 2𝑥 3 − 𝑥    𝑎𝑚𝑑  𝑧 = determine: 2 D4 DV DV (1) (2) (3) D8 D4 D8 3. Determine the following 3.1. 𝐷8 3𝑥 + 4 D 3.2. 𝑥 𝑥−2 D8 M8 N 6X 8 5 :M 3.3. 𝐷8 8 8

EQUATION OF A TANGENT AND GRAPHS Steps to find “EQUATION OF A TANGENT”: •   Find the derivate of a function ,ie, f’(x), •   Then substitute the x value given into the derivative. •   Substitute the x value into the original function equation to get the corresponding y value. •   Substitute the x and y values and the gradient in the straight line formula: y = mx + c or y – y1 = m(x – x2). PRACTICE EXERCISES ON TANGENT AND GRAPHS QUESTION 1 1.1 Determine the equation of the tangent to the curve 𝒇 𝒙 = −𝒙𝟐     + 𝟑𝒙 which is parallel to the line 𝒚 = 𝒙 + 𝟐 1.2 Find the value of p if 𝒚 = 𝒑 − 𝟗𝒙 is a tangent to 𝒚 = −𝒙𝟑       + 𝟑𝒙 − 𝟐 1.3 The graph of 𝒇 𝒙 = 𝒙𝟐     + 𝟔 has a tangent at 𝒙 = 𝒂. The tangent passes through the point (2;1) . determine the equation of the tangent. 9

Question2 2.   Below is the graph y = x 2 and the tangent to this graph at x = 3. A   0   X   3   2.1 Write down the coordinates of A. (2) 2.2 Find the gradient of a tangent at A. (2) 2.3 Find the coordinates of the point on this graph where the gradient is – 6. (2) [15] QUESTION 3 3.1 Consider the equation f ( x ) = −2 x 2 3.1.1 What is the gradient of f at x = −1? (2) 3.1.2 What is the average gradient of f between x = −1 and x = 3? (4) 3.2 Given: f : x → − x 3 − 3x 2 − 3x + 7 . 3.2.1 Determine the gradient of the curve of f at the point where x = 1. (4) 3.2.3 Hence, determine the equation of the tangent to t curve at that point. (4) 3.2.3 Determine the coordinates of the point(s) at which this tangent cuts the curve again (1) [15] 10

QUESTION 4 4. The tangent to the curve of g(x) = 2x3 + px2 + qx – 7 at x = 1 has the equation y = 5x – 8. 4.1 Show that (1; −3) is the point of contact of the tangent to the graph. (2) 4.2 Hence or otherwise, calculate the values of p and q. (6) QUESTION 5 5.1 Consider the equation  𝑓 𝑥 = −2𝑥 S + 𝑥 2 + 8𝑥 − 4 5.1.1 Determine the coordinates of the intercepts of f. 5.1.2 Determine the coordinates of turning point (stationery points) of f 5.1.3 Draw a neat sketch graph of g(x) clearly show all your intercepts with the axes and the coordinates of the turning points. 5.1.4 Determine the coordinates of the turning points of g if: 5.1.4.1 𝑔 𝑥 = −𝑓(𝑥) 5.1.4.2 𝑔 𝑥 = 𝑓(3𝑥) 5.1.4.3 𝑔 𝑥 = 𝑓 𝑥 − 2 + 2 11

QUESTION 6 6.1 The figure shows the graph of f where f(x) = −x3 + 12x. The x-coordinate of A is 1. Y   A   C     D   E   X   Determine: B   6.1.1 The y-coordinate of A. (1) 6.1.2 The equation of the tangent to the curve of f at A. (4) 6.1.3 The co-ordinates of the turning points B and C of the curve f. (6) 6.1.4 The co-ordinates of the x-intercepts D and E of the curve of f. (4) 12

6.2 The function 𝑦 = ℎ′(𝑥) is represented by the parabola in the accompanying diagram with intercept A(-2;0) B(4;0) C(0;-8). The function 𝑦 = ℎ 𝑥 is of the third degree y A(-2;0) B(4;0) x C(0;-8) 6.2.1 Write down and classify x values of the turning point of f (4) 6.2.2 For what values of x is 𝑓(𝑥) strictly increasing? (2) 6.2.3 If 𝑓 𝑥 = 𝑎𝑥 S + 𝑏𝑥 2 + 𝑐𝑥, find numerical values of a,b,c (6) 13

Y QUESTION 7 1 -1 X Consider the graph of f ’(x) shown above. 7.1 Explain why f(x) is a quadratic function (having the form y= ax2 + bx + c) (2) 7.2 What is the value of f ’(-1)? (1) 7.3 Use the graph to solve the inequalities: (a) f ‘(x) < 0 (1) (b) f ‘(x) > 0 (1) 7.4 For which values of x is f(x) increasing and decreasing? (2) 7.5 Does f(x) have a minimum or maximum turning point? Justify your answer. (2) 7.6 Find the equation of f(x). (2) [11] 14

QUESTION 8 The curve with the equation y = ax + 2x + c has a gradient of −4 at the 2 8.1 point (1; 8) on the curve. Determine the value of a and c (6) 15

9.2 The function f(x) = −2x + ax + bx + c is sketched below. 3 2 The turning points of the graph of f are T (2; – 9) and S (5; 18).   S  (5;;  18)   T  (2;;  -­9)   9.2.1 Show that a = 21, b = – 60 and c = 43. (7) 9.2.2 Determine an equation of the tangent to the graph of f at x =  1. (5) 9.2.3 Determine the x-value at which the graph of f has a point of inflection. (2) 9.3 Given 𝑓 𝑥 = 𝑥 S + 𝑥 2 + 4𝑥 + 12 Show that f is always increasing 1 9.4 Prove that the curve of y = 3 is decreasing for all real values of x , with the exception of one value for x . Give this one value for x in x your proof. 16

CUBIC EQUATIONS AND GRAPHS • Use the factor theorem to find the first factor of a cubic expression. • Use long division, synthetic division or the inspection method to find the other (quadratic) factor. • Factorize the quadratic factor into two linear factors. • Write down the three roots (solutions) of the cubic equation. CUBIC EQUATIONS •   Solving a cubic equation, e.g. 2𝑥 S + 𝑥 2 − 13𝑥 + 6 = 0: o   Use the factor theorem to find the first factor: By trial and error: 𝑓 2 = 0. Therefore 𝑥 − 2 is a factor of 2𝑥 S + 𝑥 2 − 13𝑥 + 6. o   So: 2𝑥 S + 𝑥 2 − 13𝑥 + 6 = 𝑥 − 2 … … … … o   To find the expression in the second bracket, one of three methods could be used: ü   Long division – not recommended ü   Synthetic division ü   Inspection method – recommended The result: 2𝑥 S + 𝑥 2 − 13𝑥 + 6 = 𝑥 − 2 2𝑥 2 + 5𝑥 − 3 o   The quadratic factor is then further factorised: 2𝑥 2 + 5𝑥 − 3 = 2𝑥 − 1 𝑥 + 3 o   And now we can solve the cubic equation: o   2𝑥 S + 𝑥 2 − 13𝑥 + 6 = 0 1 𝑥 − 2 2𝑥 − 1 𝑥 + 3 = 0                                                                               ∴ 𝑥 = 2;   ;  −3 2 17

GRAPHS OF CUBIC FUNCTIONS To draw the graph: • Use factor theorem and find x-intercepts. • Find y-intercept. • Calculate the turning points, using the first derivative. • Find the point of inflection using second derivative. • Now use all the above information and draw a sketch graph. •   Steps in sketching the graph of a cubic function 𝑦 = 𝑎𝑥 S + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑: o   If 𝑎 > 0, the shape will be and if 𝑎 < 0, the shape will be o   Find y-intercept: let 𝑥 = 0 and solve for y. o   Find x-intercepts: let 𝑦 = 0 and solve for x. 18

o   Stationary points: ü   Determine𝑓 C 𝑥 , equate it to zero, and solve for x. ü   Substitute the x-values of the stationary points into the original equation to obtain the corresponding y-values. ü   If the function has two stationary points, establish whether they are maximum or minimum turning points, by referring to the shape (𝑎 > 0 or 𝑎 < 0). o   Point of inflection: ü   Determine the second derivative𝑓 CC 𝑥 , equate it to zero and solve for x. Substitute the x-value of the point of inflection into the original equation to obtain the corresponding y-value. •   For all x such that 𝑓 CC 𝑥 > 0, a graph will be concave up and for all x such that 𝑓 CC 𝑥 < 0, a graph will be concave down. •   At the point of inflection of a cubic function the concavity changes. Given a graph of a cubic function 𝑓 𝑥 , learners should be able to sketch the graphs of its first derivative 𝑓 C 𝑥 and its second derivative 𝑓 CC 𝑥 on the same system of axes. They should be able to do this even if the equation of 𝑓 𝑥 was not given. Emphasize more on the relationship between cubic function to derivative function(parabola) to second derivative (straight line) Drill learners on the use of inequalities Properties of a cubic function 19

Framework  4:  Optimization  and  rate  of  change     Term 1 Topic Calculus   Weighting Sub-topics/Clarification Optimization and rate of change Related Ø   Maxima and minima. concepts/terms/vocabulary Ø   Rate /stationary points. Prior-knowledge/ Ø   Measurements (perimeter, area, Total Surface Area, volumes etc). Background knowledge Ø   Rules of differentiation. Resources Worksheet, textbooks, study guide, tin, cans, chalk box. Activities Methodology Ø   Revise with learners prior knowledge done in grade 10 and 11 (measurement) Ø   Remind them formulae (Perimeter, Area, TSA, and Volume) Ø   Look at the terminology that will be used in the sub-topic Ø   Drill learners on the concept of maxima and minima integrating with graphs (stationery points) Ø   Drill learners on the concept of rate of change and other terms they will be using in the sub-topic for example stationery point is a turning point; gradient of a tangent is a derivative Ø   Practical activity can be performed by learners in terms of optimization, e.g.: using can, boxes and their nets etc. Ø   Theory on the calculus of motion ü   S(t) represents an equation of motion (height, distance, displacement) at time t. ü   S’(t) represents velocity or speed at time t. ü   S”(t) represents acceleration at time t. 20

Assessment Ø   Class activities Ø   Short test Ø   Home work Errors/Misconceptions/ Ø   Learners use equation of Area instead of Total Surface Area Problem Areas         21

SRectangular prisms surface area= 2X A+2xB+2xC = 2xlength x width + 2 length x height + 2 x width x height volume= length x width x height surface area=lateral area +2 area of end = 2arh + 2a? = 2ar(ht+r) Cc Volume = ah QUESTION 3 A closed rectangular carton box has a square base, with side * ©”, and height / ©”. The Total outer Surface h Area of the box is 54 cm”. (Formulae; Y=27h ; 4=2x° +4xh) x 3.1 Show that the Volume V is givenby V = oy-58 : (4) 3.2 Hence, determine the value of * for which the value of ” is a maximum. (4) [8] 22

QUESTION 4 A window frame with dimensions y = h is illustrated below. The frame consists of six smaller frames y 4.1 If 12m of material is used to make the entire frame. show that y = 42 —3h). (2) 4.2 Show that the area of the window is given by A=3h - (3) 43 rina 4 and hence the dimensions, /: and _y . of the frame so that the area of the window ui isa maximum. (5) A E B QUESTION 5 ABCD is a square of side 10 em. H F Square EFGH is formed by joining the midpoints of the sides, as shown. The midpoints of those sides are joined to forma 38 square, and so on. = 5.1 Show that the sides of square EFGH are each V 50 om long.(2) 5.2 Show that the areas of successive squares form a geometric sequence and write down the 1* 3 terms. (4) (c) Calculate the sum of the areas of all the squares formed in this way. (3) [9] 23

QUESTION 8 A rectangular container has a square base and contains | litre of milk. The container has a folding lid that covers 3 times the area of the base. The length of the side of the square base is xm and the height of the container is cm. [ 1 litre = 1 000 cm? ] (a) Show that the area, A. of the cardboard used to make the container can be expressed 4000 as: Aa4x+ x ( Hint: The volume of the container = area of the base = height) (5) (b) Determine the dimensions of the container in cm, so that the quantity of cardboard used is as small as possible. (6) (11) 24

Optimization  and  Rate  of  Change  Exemplars   QUESTION 1 Car A and car B are moving along parallel straight lines of a highway. Their positions at the time t are given by: SA = 30 t 2 + 20 t + 40 ; SB = 10 t 2 + 80 t where SA and SB are the distances, in km, of car A and car B from a fixed position on the highway.              A             B       25

1.1.1 How far is car A ahead of car B when t = 0? (2) 1.1.2 When are the cars next to each other? (3) 1.1.3 When are the cars travelling at the same speed? Which car is ahead at this instant in time? (5) 1.2 [10] A farmer has a piece of land in the shape of a right-angled triangle OMN, as shown in the figure below. He allocates a rectangular piece of land PTOR to his daughter, giving her the freedom to choose P anywhere along the boundary MN. Let OM = a, ON = b and P(x ; y) be any point on MN. y N  (0  ;  b)   P(x  ;  y)   T   x O   R   M  (a  ;  0)   26

1.2.1 Determine an equation of MN in terms of a and b. (2) 1.2.2 Prove that the daughter's land will have a maximum area if she chooses P at the midpoint of MN. (6) [8] 1.3 PQRS is a rectangle formed according to the following conditions: It is bounded by the lines 𝑥 = 6 and 𝑦 = 12. Q   R(  6  ;  12  )   y=12   R is the point ( 6 ; 12 ). P lies on the curve with equation 𝑦 = and R (1;8)   8 between the points ( 1 ; 8 ) and ( 4 ; 2 ). S   R P  (  x  ;   )   8 R (  4  ;  2  )   y  =   8 x   O   x  =  6   27

1.3.1 Express the lengths of PS and RS in terms of x, the x- co-ordinate of P. (2) 1.3.2 Hence, show that the area, A square units, of PQRS is given by OR 𝐴 = 80 − 12𝑥 − (2) 8 1.3.3 Find the maximum possible area of rectangle PQRS. (5) QUESTION 2 It is estimated that the circulation of the magazine “Surfer Girl” can be represented by the equation: M (t ) = 50t 3 − 300t + 4000 , where t is the number of years of publication. (a) How many magazines were sold at the time of the initial publication? (1) (b) Derive an equation for the rate at which the circulation will change in t years. (1) (c) At what rate will the circulation be changing in 3 years time (1) (d) Determine the average rate of change of the circulation during the second year of publication, that is between t = 1 and t = 2. (3) (e) When will the magazine have the least sales? (2) [8 marks] 28

QUESTION 3 During an experiment, the volume of water in a dam is found to be V (t ) = 60 + 8t − 3t 2 , where V (t ) represents the volume of water in the dam, in kilolitres, t days after the experiment has begun. There is a suspected leak in the dam wall. a)   Calculate the rate of change of volume of water after 3 days. (3) b)   When did the volume of water start decreasing? (3) c)   When will the dam be empty? (4) 10  marks   29

Term  1   Topic   ANALYTICAL  GEOMETRY   Weighting   37  –  43  marks      (±40)   Sub-­‐topics/Clarification   1.   EQUATION OF A CIRCLE: 𝑥2 − 𝑥0 2 + 𝑦2 − 𝑦0 2 = 𝑟 2 •   Use the same formula no matter the centre is at the origin •   A point lie inside, outside or on the circle •   Completing the square to find centre 2.   INTERSECTION OF CIRCLES •   Intersect at one point: the distance between the centres is equal to the sum of their radii •   Intersection of two circles: the distance between the centres is less than the sum of their radii •   Never intersect. The distance between the centres is greater than the sum of their radii 3.   STRAIGHT LINES AND TANGENTS TO CIRCLES •   Radius/diameter is perpendicular to the tangent. •   In a tangent the product of the gradient and radius/diameter is negative one. •   Point or points of intersection of straight lines and the circle. •   The two tangents from the common point outside the circle are equal. 30

45 647 Related   Ø   Gradient  of  a  line:  𝑚 =       85 687 concepts/terms/vocabulary   Ø   Inclination  :  tanα  =  m     Ø   Midpoint:  When  given  two  points  𝑚 = 𝑥1   𝑥0 + 𝑥2 𝑦0 + 𝑦2 𝑤ℎ𝑒𝑛  𝑔𝑖𝑣𝑒𝑛  𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡  𝑎𝑛𝑑  𝑎  𝑝𝑜𝑖𝑛𝑡  𝑢𝑠𝑒 ∶ 𝑥 = ;𝑦 =   2 2 Ø   Equation  of  a  straight  line  :  𝑦 = 𝑚𝑥 + 𝑐  𝑜𝑟  𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0 )     Ø   Parallel  lines:  equal  gradients     Ø   Perpendicular  lines:  𝑚0 ×𝑚2 = −1   Ø   Collinear  points  gradients  between  the  points  are  equal.   Ø   Equation of a tangent: 𝑦 = 𝑚𝑥 + 𝑐  𝑜𝑟  𝑦 − 𝑦0 = 𝑚(𝑥 − 𝑥0 )   Prior-­‐knowledge/   Revision  of  grades  10  and  11:     Ø   Grade  10  and  11  Analytical  Geometry.   Background  knowledge   Ø   Properties  and  definitions  of  geometric  shapes.     Ø   Different  types  of  triangles.     4.   Definition:    circle,  Radius,  Tangent,  Centre   5.   Distance  formula   6.   Interpretation   of   circles(   centre   theorem;   rad-­‐tan   theorem;   tan-­‐chord   theorem)     7.   Finding  equations  of  the  straight  lines  and  tangents/normal  and  tangent   8.   Intersecting  circles     31

Resources   Ø   Chalkboard Instruments Ø   Mathematical instruments Ø   Scientific Calculator Activities     Ø   Exercises on different maths textbooks can be used Ø   Maths Handbook and Study Guide by Kevin Smith (page 221- 224). See the activities bedlow. Ø   Give leaners past examination papers   Methodology   Ø   Revise  grades  10  and  11  analytical  Geometry.  Then  give  learners  the   revision  exercises  below   Ø   Use  the  distance  formula  to  derive  the  standard  form  of  equation  of   a  circle:   (x  –  a)2  +  (y  –  b)2  =  r2  ,  where  (a  ;  b)  is  the  centre  point  of  circle  and  r   is  the  length  of  the  radius.     Ø   Then do a couple of examples from any CAPS approved textbooks Assessment   o   The practice exercises below can be taken from any relevant textbooks. •   Classwork •   Homework •   Tutorials •   Assignment •   Test   32

1  Properties  and  definitions  of  quadrilaterals,  triangles,                  circle  geometry  and  Pythagoras.  Include  definitions               2  Distance  formula                               33

§ Equation of lines 3.1 Write equation ax+by+c=0 in y- form and vice versa. 5.2 Reinforce that tamgents are straicht lmes. Tm | = rads. 3.3 Find srediant of! and | 5 lmes when one equation is given. L ANALYTICAL GEOMETRY GRADE 12 6. Equation ofcircles - ; 2 6.1 Formulas - Write in centre radius form or general fori Fieri wee sn Pl and P2 E Jes with cakbulus Euclid $-2. Use (0.0) os thecente as well at any other point as cont transi ion, alzebra, simul: ion (wt 6.3 Circle touches ~axis or axis at one point, radi " 5 a 5 equati , on ¥ value andX value respectively. finding the coordinates ofthe point ofintersection ofthe circle andthe 64 Equations of tmgents to a circle. Reinforce Tan 1 5 cadi__ Mw URES IER uae s pun su wy yrucusn uae puu nt! isosceles trianele imside Or outsidethe circle. tancents o 34

2.1 2.2 2.3 Exercise A(-1:6), B, C(3:6) and D are the vertices of a square, ABCD. B and D both lie on the line x= 1. Determine the coordinates of B and D. (6) Quadrilateral ABCD is made up by the points A(8:a). B(6:5), C(3:6) and D(2:3). If AB = AD. calculate the value of a. (4) Prove; AC 1 BD. @) Determine whether quadrilateral ABCD is a kite. Provide reasons for your answer. (5) Triangle ABC is formed by A(—1;-2), B(0:5) and C(4:1). y B(0;5) C431) + cee Px A(~1;-2) 35

Determine the midpoint of A and B, point D. Q) Determine the equation of the line passing through D and C. (3) Determine the equation of the median of triangle ABC. passing through point B_ G) Determine the coordinates of the centroid, E. @) Show that CE-ED = 2:1 (4) A(-+4:-3), B(-3:1) and C(4:2) are three points in the Cartesian plane, where AB || DC and BC || AD. C(4;2)

4.1 42 43 44 4.5 Find the coordinates of D. Determine the equation of AC. Find the midpoint of BD, point E. Prove that E lies on the straight line through AC. Determine whether ABCD is a rhombus. G) G) Q) Q) G3) 37

5.1 5.2 53 54 5.5 5.6 5. 5.1 5.2 ABCD is a parallelogram, where points A(2:5), B(5:3) and C(0;7) are given. Point D lies in the second quadrant. Ifthe gradient of BC is : , calculate the value of r. Calculate the coordinates of D. Calculate the size of angle BCD rounded off to 2 decimal places. Determine the length of BC. Determine the equation of the line perpendicular to BC, passing through point A. Determine the area of parallelogram ABCD. The points A(5:16) and B(—1:0) are the endpoints of the diameter of a circle. Determine the equation of the circle. Prove that the tangents passing through A and B are parallel to one another. 3) @) G) (2) 4) (6) (4) (4) 38

71 72 A circle has the equation x“ + y* —4x—6y +8 =0. AB is a chord with D, the midpoint of the line segment AB, being (1;3). y A Find the coordinates of the centre of the circle, as well as the length of the radius. Find the length of the line segment AB. (4) 6) 39

8.1 8.2 8.3 The points A(5;2) and B(6;-1) lie on a circle where the centre of the circle lies on the line y=2x-3. y r y=2e-3 A(5;2) Dy. + ar > B6;-1) v Find the coordinates of the centre of the circle. (6) Find the equation of the circle. (3) Find the equation of the tangent line passing through the point A. (6) 40

9.1 9.2 9.3 10.1 10.2 10.3 A circle passes through the points A(7;2) and B(3:6). The centre of the circle, M(x). is on the line 3y -—x=1. Determine the coordinates of the centre of the circle. (6) Determine the equation of the circle. (G3) Determine the equation of the tangent line passing through point B. (4) Prove that the straight line 2y— 3x =—25 is a tangent to the circle x? +y?-4x+6y =0 at B(S:-5). (6) If the point AG ;27) lies on the tangent to the circle in 10.1, find the distance from A to the centre of the circle, C. (3) Hence. find the area of triangle ABC. (5) QUESTION 1 [Extracted from “Alternate paper of WC (Question 4): Sept 2016”] The line y = x — 2 cuts the circle (x + 1)? + (y + 1)? = 20 at A(3; 1)and B, where B lies in quadrant 3. C is a point on the circumference. 41

(x +1)? +(y+1)? =20 yp=x-2 A(3;1) 42

3.1 3.2 3.3 3.4 3.5 Show that the coordinates of M, the midpoint of AB, are (— 4.; 3). Determine the equation of the tangent AD. Determine the length of AM. Determine the equation of circle centre M in the form ax’ +by?+ext+dyt+e=0 Quadrilateral ACKD is one of the following: Parallelogram: Kite: Rhombus; Rectangle Which one is it? Justify your answer. a) @ @) 1c] @ [16] 43

1.1 12 1.3 14 ABCD is a quadrilateral with vertices A(— 3 :0), B(- 1 : — 3), C(2 ; -1) and D (0; 2). ob 4 y DO; 2) A(-3 : 0) id x ce :-1) B-1;-3) oO a Show that the coordinates of B are (—3; —5) (4) Determine the distance AB. leaving your answer in simplest surd form. Q) Write down the length OB and OA. () Hence determine 6, 2BCA. Round your answer to TWO decimal places. (6) [13] 44

A(O ; 5) and B(-8 : 1) are two points on the circumference of the circle centre M, in a Cartesian Plan. M lies on AB. DA isa tangent to the circle at A. The coordinates of D are(3 : -1) and the coordinates of C are (— 12 :-1). Points C and D are joined. K is the point(0 ; -7). CTD is a straight line. , A(O; 5) f B(-8; 1 C12 ;-1} ~ A T S031) ‘K(0:-7) 3.1 Show that the coordinates of M, the midpoint of AB, are (— 4. 3). 63) 3.2 Determine the equation of the tangent AD. (4) 3.3 Determine the length of AM. @) 3.4 Determine the equation of circle centre M in the form ax? +by? +ext+dv+e=0 (4) 3.5 Quadrilateral ACKD is one of the following: Parallelogram; Kite: Rhombus: Rectangle Which one is it? Justify your answer. (4) [16]

In the diagram below, BDisa tangent to the oe D circle with centre C(3;:-2) at point B(0;2). ZA 5.1 5.2 5.3 54 5.5 Ve -2) Showthat the equation of the circle is given by x7 +? -6x+4y-12=0 v (4) Determine the equations of the vertical tangents to the circle. Give the coordinates of the respective points of contact. (4) Find the length of BD and hence show that (4:5) are the coordinates of D. (6) Determine the equation of the new circle that will be formed if the circle in question 5.1 is translated four units to the left and two units upwards. (G3) Determine the inclination of line DC. @G) 46

In the accompanying figure. K(x; v). L(—2: —1) and M(4: 3) are the vertices of triangle KLM. The equation of the side KL is y— 5x —9 = 0 and that of KM is Sv + x- 19 =0. N is the midpoint of LM. TY Kei») McG : 3) N x ) L@2:-) 6.1 Calculate the coordinates of N. (2) 6.2 Show that the coordinates of K are (-1_; 4). (4) 63 Determine the equation of the line through K and N in the form y= mx +c (3) 64 Determine the gradient of the line LM. Hence, prove that KN is the perpendicular bisector of LM. GB) 6.5 IfL, M and the point J(7; a) are collinear, calculate the value of a. (3) 6.6 Determine the size of the angle of inclination between KL and the positive x-axis. (2) (17) 47

Framework  6:  Trigonometry  Revision     Term  1   Topic   TRIGONOMETRY     Weighting   27-­‐33  marks(march  test)   Sub-­‐topics/Clarification   Ø   GRADE 11 TRIGONOMETRY REVISION Ø   COMPOUND ANGLES IDENTITIES Ø   DOUBLE ANGLE IDENTITIES Related  concepts/terms/vocabulary   Ø   USE  OF  QUOTIENT  IDENTITY,     Ø   SQUARE  IDENTITY,     Ø   REDUCTION  FORMUALE,     Ø   SPECIAL  ANGLES,  CO  FUNCTIONS,     Ø   NEGATIVE  ANGLES,     Ø   RIGHT  ANGLED  TRIANGLES,     Ø   GENERAL  SOLUTION     Prior-­‐knowledge/   Ø   DEFINITION  OF  TRIG  RATIOS.   Ø   PYTHAGORAS  THEOREM.   Background  knowledge   Ø   DEFINITION  OF  QUADRANTS,   Ø   FACTORISATION     48

Resources   Ø   CALCULATOR, Ø   REVISION WORKSHEET Ø   TEXTBOOK Ø   PREVIOUS YEARS’ QUESTION PAPERS Activities   Ø   DIAGNOSTIC EXERCISE(trig grade 11) Ø   CLASSWORK – DISCOVERY EXERCISE Ø   TUTORIAL Ø   TRIG GRAPHS REVISION Methodology   Ø   Revise of grade 11 trigonometry from any textbook. •   Learners must be able to apply the theorem of Pythagoras accurately on the following type of problems: v   Learners must be able to write problems such as sin16°=t, e.g. t as a ratio in a form   𝑎 𝑏 02 v   When given 𝑠𝑖𝑛𝜃 = − , 90° ≤ 𝜃 ≤ 270; learners need to be able to identify the correct 0S quadrants. •   Learners  to  be  encouraged  to  read  and  follow  the  given  instructions  e.g.  when  told  not  to  use  a   calculator  they  won’t  be  awarded  marks  if  ignoring  that   •   Revision of basic algebra •   Learners must be familiar with the following trig identities: z{F|                                                                              𝑠𝑖𝑛𝜃 2 + 𝑐𝑜𝑠𝜃 2 = 1    and  𝑡𝑎𝑛𝜃 =     }~z| •   Learners  must  be  able  to  use  the  CAST  strategy  to  simplify.   •   When  given  negative  angles,  learners  need  to  just  add  360°  to  get  a  positive  equivalent  angle.   •   When  given  angles  greater  than  360°,  learners  need  to  just  subtract  360°  to  get  an  equivalent  angle.   •   Drill  learners  to  use  the  calculator  to  sketch  trig  graphs.   •   When  given  a  trig  graph,  learners  need  to  be  able  to  explicitly  interpret  it.   Ø   Use the activities below to deduce that: 49

•   Cos( A – B) ≠ Cos A – Cos B •   But Cos (A – B) = CosACosB – Sin ASinB Guiding Activity 1.Given (i) ∠A = 90° and ∠B = 30° (ii) ∠A = 120° and ∠B = 30° (iii) ∠A = 225° and ∠B = 135° Use a calculator to calculate each of the following: (a) cos(A – B) (b) cosA – cosB (c) cosAcosB + sin AsinB 2. What conclusion(conjecture or deduction) can you make 1 above? •   cos( A – B) = = cosACosB + sin AsinB, will then be used to derive the other compound and double angles. •   Then the teacher must demonstrate with some examples the application of the identities from any relevant textbooks. Errors/Misconceptions/   − sin x − tan x ≠     − cos x Problem  Areas   2   tan θ = ∴ x = 3 / y = 2     3 cos 2 x − sin 2 x = 1     cos(A ± B) = cosA ± cosB           sint x = t sin x cos x     50

Assessment   Ø   Classwork:  Revision  Worksheet  (2  hours)   Grade  11  Trigonometry  Revision     CALCULATORS  MAY  ONLY  BE  USED  FOR  QUESTIONS  1    AND  4.       1.    Give  answers  correct  to  2  decimal  places.         A = 1210 B = 61,20 a)    If      and    determine  the  values  of:        i)    𝒄𝒐𝒔  𝑨   +  𝒄𝒐𝒔  𝑩                   (2)      𝒊𝒊)    𝒄𝒐𝒔  (𝑨   +  𝑩)                   (2)     1 iii)      -­‐   tan B                                              (2)   tan 2 A   51

b)    If  OP  =  10  units,  calculate  the                                  y                  value  of  x.                      P  (x;  y)                                            160°                                        O                                                    x                         (3)                                            [9]     02 2a).    If  𝑡𝑎𝑛𝜃 = ;    θ    ∈  (90°;  270°),  calculate  the  values  of:   M      i)    cos  θ      ii)    sin2  θ                                   52

b)    Copy  and  complete  the  following:   sin A i)   tan  A  =     ?    ii)    1  -­‐  sin2  A    =    ….                     1    c)    Simplify:     (1  + ).  sin2  A             tan 2 A      d)    Prove,  using  fundamental  identities:             1 sin  A.  tan  A        =             -­‐    cos  A                             cos A  e)    If  sin  40    =  t,  write  in  terms  of  t:        i)    cos  50°      ii)    sin  320°      iii)    cos  40°        iv)    cos(310°)                                 53

QUESTION  3     3.1   ( ) In  the  diagram  below,   P 2 3 ; − 2  is  a  point  in  the  cartesian  plane,  with  reflex  angle     QÔP = α .    Q  is  the  point  on  the  x  –  axis  so  that   OP̂Q = 90 o           y            O   x       Q                 54

Calculate  without  measuring:           3.1.1   β .                     (3)     3.1.2   the  length  of  OP.                     (2)     3.1.3   the  co-­‐ordinates  of  Q.                     (3)   3.2   If  cos   α + 3 sin α = k sin (α + β ) .       Calculate  the  values  of  k  and   β .                                          (5)                                 3.3    In  the  diagram  alongside,   D Oˆ P = 90!.  The  angle  which  OP     makes  with  the  x  –  axis  is θ .       Given  D  (b;4)  ;  P  (6;a)  and   5 cosθ − 2 = 0   55

3.3.1   Determine,  without  the  use  of  a  calculator,     the  numerical  value  of:     3.3.1.1   a                                                                                                                                                                                                                                                                (3)       3.3.1.2   b                                                                                                                                                                                                      (4)     3.3.2   Simplify  without  the  use  of  the  calculator.     1.         sin 2 20 ! + sin 2 70 !                 (3)     cos 330! .sin 140!   3.3.2.2                                        (10)     sin (−160! ) . tan 405! .sin 290!                                                     56

3.4   Show  without  using  a  calculator  that:     6− 2   sin 15o =                     (3)   4 3.5   Write  the  following  as  a  single  trigonometric  ratio:   tan 180 o + θ . cos 360 o − θ ( ) ( )   ( sin 180 o − θ . cos 90 o + θ + cos 540 o + θ . cos (− θ ) )       ( ) ( )       (9)     3.6   Prove  that:     sin 45 + x . sin 45 − x = ( o ) ( o ) 1 2 cos 2 x           (5)   sin 33 o cos 33 o 3.7   Show       −    =    2                 (6)   sin 11o cos 11o tan 3x 3.8   Determine  the  general  solution  of  the  equation:     − 1 = 0         (5)   tan 24° 3.9                Prove  the  identity:   (sin θ − cos θ ) 2 1   = − 2 tan θ                                                                                                                                                                                                                  (5)   1 − sin θ 2 cos 2 θ 57

3.10     1   3.10.1   If cos( x + 30°) = −2 sin x, deduce that tan x = −                                                                                                   (5)   3   3.10.2   Hence  determine  the  general  solution  of  cos  (x    +    30o)    =    –2  sin  x.       (2)   4.    Simplify:   tan(180° − A) sin(180° + A) cos(180° − A) a)                                                                                                    (8)   cos(360° − A) tan(360° − A)   b)    sin60°.  tan30°    +    cos  60°.  tan  45°                                                                                                (6)   2 ⎛ 1 ⎞ c)     ⎜ cos 210 °. ⎟                                                                                                  (5)   ⎝ sin 225 ° ⎠ d)    sin2  80°  +  cos2  260°                                                                                                    (2)   cos 330°. sin160° e)                                                                      (8)   tan 405°. tan( −300°).sin( −160°)                                                                                                                                                                                                                                                                                                                                                                                  [29]       58

5.   If  a  =  250,4°  and  b  =  84,3°,  use  a  calculator  to  evaluate,  correct  to  2     decimal  places:   ⎛a ⎞ a)   sin   ⎜ + b ⎟   ⎝2 ⎠ b)   tan  (a  -­‐  b)   c)   ½    cos  2b                                        [6]   12 6.   If  sin  A    =    -­‐      and   Â ∈ [90°  ;;  270 °] ,  use  a  diagram  to  determine     13 5 12 the  value  of   + .                          [6]   cos A tan A     sin 240 °. tan 210 °. sin 20° 7.   Evaluate:                          [7]   sin 315 °. cos 135 °. cos 110 °   59

8.    Simplify:       sin(180° − x).sin(180° + x) a)                                    (7)   cos(90° + x). tan( x − 360°) tan(180° + x)                                                                     9.   Prove:     2 sin x 1    =    cos2  x  +  sin2  x    -­‐   2 −  tan2  x       [6]   tan x. cos x cos x       10.    Simplify,  using  fundamental  identities:     a)    1  –  sin2  x  –  cos2  x                 (2)   60

b)    sin4  x  –  sin2  x.  cos2  x                 (3)     1 1 c)     +               (4)   1 + sin A 1 − sin A   sin B 1 d)     +                 (4)   1 + cos B tan B   11.  a)    α  and  β  are  complementary  acute  angles  and  5  tan  α  =  12.   5 1                  Evaluate − .               (5)   sin β tan α        b)    P  is  the  point  (-­‐1;  b),   XÔP  =  θ  and  180°  <  θ  <  270°.                      Determine  the  value  of  b,  if  θ  =  210°.         (3)         61

Compound  Angle  worksheet   Give  answers  correct  to  2  decimal  places,  where  necessary.   1.    Evaluate  the  following  without  using  a  calculator:   a)    cos(70°  -­‐  x)cos(x  -­‐  10°)  –  sin(70°  -­‐  x)sin(x  -­‐  10°)   b)    sin75°cos75°   c)    cos(-­‐  52°)cos22°  +  cos38°.cos292°   2 sin 144° cos( −36°) d)       tan 126°. cos 234°. sin 324° e)    cos64°  +sin64°.tan32°   sin15°. cos15° f)       sin 30°. cos 45° sin( x + 300°) g)       cos(150° − x ) 62

cos10° cos 340° − sin 190° sin( −380°) h)       sin 80° cos 20° + cos 820° cos 70° QUESTION  2   THIS  QUESTION  HAS  TO  BE  ANSWERED  WITHOUT  THE  USE  OF  A  CALCULATOR:   †‡ˆ0O‰°.Š‹ˆ  (6S0M°) 2.1   Simplify  fully:     2.1.1             (5)   Œ•†2S‰°.†‡ˆO2‰° †‡ˆ0M°.Œ•†0M°         2.1.2         (5)   Œ•† OM°68 Œ•†86†‡ˆ OM°68 †‡ˆ8 2.2.1   Express  cos 2 𝐴      in  terms  of  cos2A               (2)   S:2 2.2.2   Hence  show  that  cos15°  =                   (4)   2 2.3   Calculate  𝑥  when  sin2𝑥  =  cos(−3𝑥)  for  𝑥 ∈ [−90°; 90°]                                    (6)                     :                                                                              [22]                           2.4    If  cos  40°  =  t,  express  each  of  the  following  in  terms  of  t                    2.4.1    tan  50°                    2.4.2    cos  80°   63

2.4.3    cos  70°   5 3 3a)    If  sinα  =  -­‐    ;  tan  α  <  0  and  cosβ  =   ;  180°  <  β  <  360°,  evaluate:   13 5                  i)    cos(α  -­‐  β)                  ii)    sin  2α                iii)    cos(180°  -­‐  2β)     24      b)    If  tanA  =  -­‐ ;  90°  <  A  <  270  evaluate:   7                  i)    sin  2A                  ii)    cos  2A     4.    Prove:   a)  i)    cos(A  +  B)  +  cos(A  –  B)    =    2cosA  cosB                                    ii)      Hence  show  that  cos80°  +  cos  40  =  cos  20°  (no  calculator)   1 − cos 2x     b)     = tan 2 x   1 + cos 2x 64

tan x + sin x x     c)       = cos 2   2 tan x 2     d)    cos4  A  –  sin4A    =  cos2A   cos 2x + cos x 1 + cos x     e)     =   sin 2x − sin x sin x. 1 + tan x     f)      tan(45°  +  x)    =       1 − tan x sin x + cos x     g)    sin(x+45°)    =       2 cos( −2x ) + cos 2 x + 3 sin 2 x 1     h)     =   2 − 2 sin x2 cos 2 x           65

Double  angle  identities   sin 2𝜃 = 2 sin 𝜃 cos 𝜃   cos 2𝜃 = 𝑐𝑜𝑠 2 𝜃 − 𝑠𝑖𝑛2 𝜃   = 1 − 2𝑠𝑖𝑛2 𝜃   = 2𝑐𝑜𝑠 2 𝜃 − 1   Half  angles(  only  for  enrichment)   𝜃 1 − cos 𝜃 sin =±   2 2 𝜃 1 + cos 𝜃 cos =±   2 2 sin 5 x + sin x (b)   (i)   Prove:   = sin 3x     2cos 2 x     (Hint:   5x = 3x + 2x and x = 3x − 2x )                                (5)   Prove:   2sin 2 x 1 =   2 tan x − sin 2 x tan x 66

1.2              Prove  the  following  identity:   cos A  -­‐   cos 2A  +    2 1  +   cos A      =                                           (6)     3 sin A-­‐ sin 2A   sin A 5.1   Simplify,  without  the  use  of  a  calculator:   †‡ˆ 0O‰°.†‡ˆ 02‰°                                                                                                                                                        (6)                                                                                   †‡ˆ 00‰°.†‡ˆ SO‰° 5.2   Given:       †‡ˆ 8:†‡ˆ 28                                       = tan 𝑥         0:Œ•† 8:Œ•† 28 5.2.1   Prove  the  above  identity.                                                                                                                                                                                (4)                                                                                                                                                                     67

Trig  ratios  –Double  angles   It  is  given  that  𝑐𝑜𝑠  27° = 𝑘.  Use  a  sketch  to     express  each  of  the  following  in  terms  of  𝑘 ,     clearly  showing  all  working  detail:     1.1.1   𝑐𝑜𝑠²(−27°)                                                                                                                      (2)     1.1.2   𝑠𝑖𝑛  423°                             (3)     1.1.3   𝑠𝑖𝑛  27°                                     (2)     1.1.4   𝑠𝑖𝑛  54°                                   (3)   3     4.1   Given:   sin α = and 90o < α < 270o   5   With  the  aid  of  a  sketch  and  without  using  a  calculator,  determine:         4.1.1   tan α          (3)         4.1.2   cos (90 + α )     (2)     4.1.3   cos 2α     (3)   68

Given:   sin 38o = p ,  determine  the  following  in  terms  of  p.       4.2   (  Without  using  a  calculator.)     4.2.1   ( cos −38o   )   (3)     4.2.2   sin 76o     (2)     (a)   Refer to the sketch below. (b)   θ ˆ with B (4; –3) given. is the reflex angle XOB   sin 2θ   Determine,  without  the  use  of  a  calculator,  the  value  of   .                                                                                        (3)   cos θ   69

0M 4.1   Given:  sin 2𝐴 =  for  0° ≤ 2𝐴 ≤ 90°       R 4.1.1   Determine  the  value  of  cos 2𝐴  without  the  use  of  a  calculator.                                                                                                  (3)     4.1.2   Hence,  find  the  value  of  cos 𝐴  without  the  use  of  a  calculator.  Leave  your  answer  in  surd  form.           (4)   K   (3D  problems)       α                                                                           2                                   L                                                              90°  +  α                                                   M           2α     N    In  the  sketch,  KL  is  a  vertical  tower  and  L,  M  and  N  are  all  points  in  the  horizontal  plane.     ML̂N = (90 ° + α ) ;   N̂ = 2α and K̂ = α.    KM  =  2  units.    a)    Express  ML  in  terms  of  a  trigonometric  ratio  of  α  and  hence  show  that                MN  =  1  unit.                     (4)     70

b)    Prove,  using  the  sine  rule,  that                      LN  =   1  –  4  sin2α                          (HINT:  cos3α=  4cos3α  –  3cosα)                                                                                            (5)     In   the   diagram   alongside,   MT   is   a   vertical   M structure.   P,  Q  and  T  are  three  points  in  the  same  horizontal   plane.   The  angle  of  elevation  of  M  from  Q  is   θ .   PQ  =  k  metres.   ˆ = 2θ   PM  =  2  PQ        ,           MPQ P 2θ   T 1 2 area Δ MPQ = k 2 ⋅ sin θ ⋅ cos θ   k ϴ Q   17.1   Hence,  or  otherwise,  prove  that     MQ = k. 1 + 8sin 2 θ   (5)     71

17.2   Find  the  value  of  MQ,  rounded  off  to  the  nearest  metre  if   k = 139,5 m  and   θ = 42o .                               (2)   2ab 5a)    In  Δ  ABC   Ĉ = 90°.  Prove  that  sin2A    =       c2    b)    Show  that  sin(x  +  40°).cos(x  +  30)    -­‐    cos(x  +  40°).sin(x  +  30°)  is                  independent  of  x.      c)    If  cos2A  =  -­‐  cos2B,  prove  that  cos2  A  +  cos2  B  =  1        d)    If  sinx.  cosx    =    k,  express  sinx  +  cosx  in  terms  of  k     6.    Solve  for  x:   a)    10sin2  x  +  cosx  –  7    =    0;    x  ∈  [-­‐180°;180°]     b)    sin5x.cos3x    -­‐    cos5x.sin3x    =    cosx  ;    x  ∈  [-­‐90°;90°]   c)    sinx.  sin2x  +  cos2x    =    1   d)    sin3x  –  cos2x    =    0   72

e)    cos54°.cosx  +  sin54°.sinx    =    sin2x   f)    cos70°.cos10°  +  sin70°.sin10°    =    2cos2  2x  –  1   g)    sin2x    =    -­‐  cosx   h)    sin2x  +  sin2x    =    0             (sin x − cos x ) 2 − 1 i)        =    2   sin 2 x − 1     73

TRIGONOMETRY GRAPH WORKSHEET QUESTION 1 State  the  maximum  value,  the  minimum  value  and  the  period  of  the  following  functions:     1.1   y  =  -­‐  2½  sin  3x   (3)     1.2   y  =  5  cos  x  -­‐  3   (3)     1.3   y  =    -­‐3  sin  (x  –  60o  )                                                                                                                                                               (3)       [9]   QUESTION  2     Give the equations of the following graphs: 2.1                         74

y (60o;;  1) x 0° 90° 180° 270° 360° (240o;;  -­1) 2.2                                                                                   (2)                                                                                           2.3                         75

2.4           y 3 2 1 x 0 22,5o 45° 90° 135° 180° -­1 -­2 -­3   (2)                                 76

QUESTION  3     Draw  sketch  graphs  on  separate  sets  of  axes  of  each  of  the  following  for  the  interval    x  ε  [-­‐180o  ;  180o]       3.1   y  =  tan  x  -­‐  1   (3)     3.2   y=    cos  2x   (3)     3.3   y  =  -­‐  2  sin  x   (3)         [9]           77

QUESTION 4 y g C B f x 135o -­360° -­270° -­180° -­90° 0° 90° 180° 270° 360° A         In  the  figure  the  graphs  of    f  =  {(x;y)|  y  =  cos  (x  +  a)  }    and  g  =  {(x;y)|  y  =  tan  bx}     for  x  ∈  [-­‐360o  ;  360o]  are  given.  The  graphs  intersect  at  B  and  C.     78

4.1    Determine  the  values  of  a  and  b  by  using  the  graphs   (2)   4.2    Calculate  the  value  of  f(0°)  without  using  a  calculator,  leaving  your  answer                  in  surd  form.   (3)   4.3    Determine  the  co-­‐ordinates  of  A.   (2)   4.4    If  B  is  (79°;  0,82),  determine  the  co-­‐ordinates  of  C.   (2)   4.5    Determine  x  by  using  the  graphs  if  f(x)  ≥  g  (x)  and    x  ∈  [-­‐180°;  180°]   (2)       [11]                   79

QUESTION 5   Consider  the  functions  :   f  =  {(x;y)|  y  =  sin  (x-­‐60o)  }   g  =  {(x;y)  |  y  =  cos  x  +  0,5  }   5.1  Draw  sketch  graphs  of  f  and  g  on  the  same  set  of  axes  for  the  interval  [-­‐120o;  90o]   Indicate  all  the  intercepts  with  the  axes  as  well  as  the  co-­‐ordinates  of  the  point(s)  of  intersection  of  the  graphs   (8)   5.2  From  the  graphs  write  down  the  value(s)  of  x  for  which    f(x).  g(x)  ≥  0.   (3)   5.3  If  the  y-­‐axis  is  moved  30o  to  the  left,  which  function  will  then  be  represented  by  f?   (2)         [13]                   QUESTION  6   The  graphs  below  represent  the  functions  of  f  and  g.      f(x)  =  sin2x  and  𝑔(𝑥)  =  𝑐 sin𝑑𝑥,  𝑥 ∈ [0°; 180°]   80

g   f       6.1   Determine  the  value(s)  of  x,  for  𝑥 ∈ 0°; 180°  where:     6.1.1   𝑔(𝑥)  – 𝑓(𝑥)  =  2                 (1)     6.1.2   𝑓(𝑥)  ≤ 0                   (2)   81

6.1.3   𝑔(𝑥). 𝑓(𝑥)  ≥ 0                 (3)                      6.1.4  Determine  the  value(s)  of    x  for  which  f(x)  is  strictly  decreasing.                                          (2)     6.2   𝑓    in  the  graph  drawn  above  undergoes  transformations  to  result  in    𝑔    and    h    as  given  below.  Determine  the  values   of  𝑎 , 𝑏, 𝑐  and  𝑑    if     6.2.1   𝑔(𝑥)   =  𝑐sin𝑑𝑥                 (2)     6.2.2   ℎ(𝑥)   =  𝑎cos(𝑥 − 𝑏)                 (2)                                                                    [12]               82

Trigonometric  2D  Exemplars       TRIGONOMETRY   THE  THEOREM  OF  PYTHAGORAS     In  a  right-­‐angled  triangle,  the  square  on  the  hypotenuse  is  equal  to  the  sum  of  the  squares  on  the  other  two  sides  (right  angle   sides).     The  Theorem  of  Pythagoras:   1.  Only  applies  to  a  right-­‐angled  triangle   2.  Uses  the  lengths  of  the  sides  of  a  right-­‐angled  triangle   3.  Uses  the  squares  of  the  sides  of  a  right-­‐angled  triangle   4.  The  hypotenuse  is  opposite  the  90°  angle   5.  The  hypotenuse  is  always  the  longest  side  of  a  right-­‐angled  triangle   83

6. The General mathematic > 14,32 18,67 24 84

𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 (𝑷𝒚𝒕𝒉. 𝑩 = 𝟗𝟎°)       85

P x R z 2 2 2 e=yt+2Zz =p+¢ y SEEBG SEER) ¥=%7%, p=r—@ = y q =n —p Hypotenuse selon F pPai@+e 2 p22 a ae Hypotenuse 86

From  the  three  examples  above  we  can  see  that:   ü   If the hypotenuse has to be determined: always a + is used (𝒂𝒍𝒘𝒂𝒚𝒔  𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆)𝟐 = (𝒐𝒕𝒉𝒆𝒓  𝒔𝒊𝒅𝒆)𝟐 + (𝒐𝒕𝒉𝒆𝒓  𝒔𝒊𝒅𝒆)𝟐     ü   If a right angle side has to be determined: always a – is used (𝒐𝒕𝒉𝒆𝒓  𝒔𝒊𝒅𝒆)𝟐 = (𝒂𝒍𝒘𝒂𝒚𝒔  𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆)𝟐 − (𝒐𝒕𝒉𝒆𝒓  𝒔𝒊𝒅𝒆)𝟐       The  Theorem  of  Pythagoras              ∴  𝑏 2 = 𝑎2 + 𝑐 2  in  the  figure   Or  𝑎2 = 𝑏 2 − 𝑐 2   87

Or  𝑐 2 = 𝑏 2 − 𝑎2  by  changing  the  subject  of  the  formula     The  Converse  of  the  Theorem  of  Pythagoras     If    𝑏 2 + 𝑎2 = 𝑐 2  then  𝐶 = 90°   If    𝑎2 + 𝑐 2 = 𝑏 2  then  𝐵 = 90°   If    𝑏 2 + 𝑐 2 = 𝑎2  then  𝐴 = 90°                       88

WORKSHEET  ON  THE  TRIGONOMETRICAL  RATIOS           3.  Given  a  ∆PQR,  in  which  𝑅 = 𝜃       Determine  the  values  of:   3.1        PR  = ⋯                                                                                                                3.5            𝑠𝑖𝑛2 = ⋯     89

3.2      𝑠𝑖𝑛𝜃 =                                                                                                                    3.6          𝑐𝑜𝑠 2 = ⋯       3.3      𝑐𝑜𝑠𝜃 = ⋯                                                                                                        3.7        Hence,  determine  the  value  of  𝜃   3.4    𝑡𝑎𝑛𝜃 = ⋯       3.8  Without  using  a  calculator,  determine  the  following     z{F| 3.8.  1.         = ⋯                                                                                              3.8.2.      𝑠𝑖𝑛𝜃 2 + 𝑐𝑜𝑠𝜃 2 = ⋯     }~z| 3.8.3.      1 − 𝑠𝑖𝑛𝜃 2 = ⋯                                                                                3.8.4.          1 − 𝑐𝑜𝑠𝜃 = ⋯     3.8.5.       1 − 𝑠𝑖𝑛𝜃 1 + 𝑠𝑖𝑛𝜃 =…                                        3.8.6.           1 − 𝑐𝑜𝑠𝜃 1 + 𝑐𝑜𝑠𝜃 = ⋯                 90

4.      The  Area  Rule    Determine  the  area  of  ∆TVS       5.  Determine  the  area  of  ∆TVS  in  terms  of  𝒕     91

6.  Determine  the  size  of  the  V  and,  hence  T     The  Sine  Rule     1.   The proof of the Sine Rule must be done by the teacher and the learners (Refer to the Book, (Book work)) 92

2.   Application of the Sine Rule: Determine the length of KZ and ZN in the following triangles: Type 1 Type  2 In  the  above  triangles  find  the  length  of  BC   93

Type  3   2.6.   In the above diagram find the size of 𝑁 2.7.   In the following diagram find the value of 𝐵 2.8.   In the following diagram find the length of OP and LM THE COSINE RULE 94

Type  1     Two  sides  and  an  included  angle  given:     95

Type  2   Given  all  the  three  sides:  Determine  all  the  three  angles.      ACTIVITIES  ON  2DS  in  Grade  11   TRIGONOMETRY:    SOLVING  TRIANGLES   For  each  of  these  triangles,  find  the  side  marked  with  a  “  ?  “.   Give  your  answer  correct  to  one  decimal  place.  Diagrams  are  NOT  drawn  to  scale.     Q   .   2.   ?   A   105°   1.     97,9°     R   5   29,7°   C   30°   12     P   ?   B   96

For  each  of  these  triangles,  find  the  3rd  angle  and  then  the  side  marked  with  a  “  ?  “.  Give  your  answer2  correct  to  one  decimal  place.  Diagrams  are  NOT   drawn  to  scale.   A   Q   3.     ?   4.   6,2     105°     29,7°   C   45°   R     52,4°   10   B   ?     P   For  each  of  these  triangles,  find  the  3rd  side  AND  the  area.   Give  your  answer  correct  to  one  decimal  place.  Diagrams  are  NOT  drawn  to  scale.     Q   6.     105°   .   5.   A     8,8   8   ?   R     5   ?   C   12     P   52,4°     B   Q     8.   ?   7.     A   8   8,8   R   ?   30°   29,7°   C   12   P   97 10   B

Trigonometric 3D Exemplars                                                                                                                                                                                   1.   The  wedge  of  cheese  shown  alongside  is  called  Emmenthaler.     30°     Ignore  its  holes  when  answering  the  questions  below       The  wedge  is  a  right  prism.     The  base  of  the  prism  is  triangular  and  viewed  from  the  side.     The  hypotenuse  of  the  triangle  is  17  cm  and  the  angle  shown  is  30°.     The  other  dimension  shown  is  10  cm.     Calculate         (a)   the  surface  area  of  the  wedge  of  cheese     (b)   the  volume  of  cheese     98

STEP  1     Use  trig  ratios  in  ΔABC  to  find  AB  and  CB     30°       STEP  2   Find  the  areas  of  the  following  and  then  add  them  up  to  get  the  surface  area  of  the  wedge   •   Rectangle EFBA   •   Rectangle BCDF   •   Rectangle ACDE   •   ΔABC   •   ΔEFD STEP  3   99

Volume is base area x height = area ΔABC x DC. Calculate the volume of the wedge     3.   The  vertical  flag  pole  in  the  picture  is  8  m  high.     The  two  stays  that  keep  the  pole  upright  are  10  and  17  m  long.     The  distance  between  the  bases  of  the  two  stays  is  12  m.     Find  the  area  of  the  triangle  formed  by  the  bases  of  the  pole       and  the  two  stays.     Show  all  your  reasoning  and  working.     You  will  need  to  use  Pythagoras,  the  cosine  rule  and  the  area  rule.         STEP  1   Use  Pythagoras  in  ΔPQS  to  find  QS   STEP  2   Use  Pythagoras  in  ΔPQR  to  find  QR   100

STEP  3   Use  the  cosine  rule  in  ΔQSR  to  find  angle  Q   STEP  4   Use  the  area  rule  in  ΔQSR  to  find  the  area  of  the  shaded  triangle     A       2. B, C and D are points in the same horizontal plane. AB is perpendicular to this horizontal plane. The angle of elevation of A from C is 43°. CD = 15 cm ˆ = 40° ˆ = 30° B   CBD CDB     C   D   Find the height of A above B.     STEP1     Add  all  the  given  measurements  to  the  diagram     101

STEP  2   Use  the  sine  rule  to  find  BC  in  ΔABD   STEP  3   Find  AB  in  right-­‐angled  ΔABC     4.   In  the  diagram,  two  men  labelled  A  and  C,  are  standing  in  the  same  horizontal  plane  as     the  foot  F  of  a  building.  The  height      of  the  building,  FB,  is   28  metres.     The  distance  between  the  men,  AC,  is  50  metres       The  angle  of  elevation  of  building  B,  as  measured  from  man  A,  is  21°.       Angle  FAC  is  30°.   B     Find  the  distance  between  from  man  C  to  the  foot  of  the  building  F.     28     A   21°     30°   F   50   STEP  1   C   B   Shade  the  horizontal  plane   STEP  2   28   Find  AF  in  right-­‐angled  ΔABF     STEP  3   A   21°   30°   F   50   C   102

Use  the  cosine  rule  to  find  CF  in  ΔACF       5.   This  is  the  2012  NSC  Paper  2  question  on  “solving  triangles  and  3-­‐d.”                             103

STEP  1   Re-­‐draw  the  diagram  and  shade  the  horizontal  plane     STEP  1   Use  the  sine  rule  in  ΔCBD  to  find  CB     STEP  2   Use  the  Pythagoras  method  in  ΔHCB  to  find  HC     STEP  3   Use  the  cosine  rule  in  ΔHCD  to  find  angle  θ     Remember:   If  you  have  three  measurements  for  a  triangle,  you  can  find  all  the  others  .  .  .     6.     March  2013  Paper  2  Common  Test   R   O  is  the  centre  of  the  semi-­‐circle  PRQ  and    r    is  the  radius.   PO  is  the  diameter  of  the  semi-­‐circle  PO.   O x   P Q ˆ = 90°.   It  is  further  given  that   Q̂ = x  and   PTO   T 104

Determine       6.1   RQ  in  terms  of    r    and    x                                                                                                                                            (6)     6.2   the  area  of  ΔROQ  in  terms  of    r    and    x                                                                                                                                        (3)   6.3   the  area  of  ΔPOT  in  terms  of    r    and    x                                                                                                                                          (6)     area of ΔROQ 6.4   the  ratio      if     x = 75° .                                                                                                            (4)   area of ΔPOT   Leave  your  answer  in  simplified  surd  form.         STEP  1   Write  an  ‘r’’  on  all  the  line  segments  equal  to  the  radius       Fill  in  all  the  angles  in  terms  of    x     STEP  2   Use  the  cosine  rule  in  ΔROQ  to  find  ‘RQ  in  terms  of    r    and    x’     105

STEP  3   Use  the  area  rule  in  ΔROQ  to  find  ‘area  of  ΔROQ  in  terms  of    r    and    x’     STEP  4   Use  trig  in  right-­‐angled  ΔPOT  to  find  OT  in  terms  of  r  and  x     STEP  5   Use  the  area  rule  in  ΔPOT  to  find  ‘area  of  ΔPOT  in  terms  of    r    and    x’     STEP  6   Make  a  fraction  from  your  answers  to  6.2  and  6.3,  substitute  75°  for  x         and  see  what  happens  .  .  .     EXERCISES A   B   1. ABCD is a rectangular sheet of paper. AB = 8cm, BC = 10cm and M is the midpoint of AD. The paper is folded on line BM such that A' is the new position of A. Calculate (a) Angle ABM (b) Angle A'BC A'   (c) The length of A'C D   C   106

2. Two identical rhombuses ABCD and EFCD are placed at right-angles against each other. A   B   ADCˆ = 60° , AD = h units and AK ⊥ DC , Calculate the following lengths in terms of h (a) AK (b) DK (c) KF K   C   D   If the two rhombuses are pressed against each other so that the angle between them is < 90° 21 h2 the area of ∆AKF becomes F   16 E   (d) Calculate the angle between the two rhombuses. 3. The cover of a book EABCDF stands upright as in the figure. A   B   AC and AF are the diagonals of identical rectangles ABCD and AEFD respectively. AB = p units and CF = q units. E   ˆ = 1− q2 (a) Show that cos CDF 2p2 (b) If p = 15 and q = 12, calculate (i) ˆ the size of CDF (ii) ˆ = 27,8° the length of AC if FAC D   C     F   107

D   4.   A,  B  and  C  are  three  points  in  the  same  horizonal  plane.     DA  is  a  vertical  cliff.     The  angle  of  elevation  from  B  to  the  top  of  the  cliff  is  θ.     ˆ = α .   ˆ = ACB ABC A     The  distance  between  B  and  C  is  k  metres.   B   C   k tan θ   Prove  that   AD =   2cos α         A   5.   B,  C  and  D  are  three  points  in  the  same  horizontal  plane.     AB  is  a  vertical  pole  of  length  p  metres.   p     The  angle  of  elevation  of  A  from  C  is  θ.   B     ˆ =θ BCD ˆ = 30° CBD BD = 8 m   30°   8       ( Prove,  without  using  a  calculator  that   p = 4 1 + 3 tan θ metres.   ) θ   D     C   108

6.   AB  is  a  vertical  corner  post  and  AP  and  AQ  are  two  cables.     P,  Q  and  B  are  in  the  same  horizontal  plane.     The  height  of  the  corner  post    AB    is    x    metres.   A     PQ  =  2x  metres     The  angle  of  elevation  of  A  from  Q  is    θ.     ˆ APQ = 90° − θ ˆ PAQ = α   P   B     Determine   α  in  terms  of  θ.   x2 Q     If    α  =  2θ    show  that  the  area  of  ∆APG  =     tan θ PROJECT   1.   Study  the  diagram  below.   2.   Read  the  written  description  of  the  diagram  carefully.   3.   Make  a  3-­‐dimensional  model  of  the  figure  in  the  diagram.                     Use  a  ruler,  scissors,  paper,  etc.  and  mark  everything  given  on  your  model.   A     B,  C  and  D  are  three  points  in  the  same  horizontal  plane.   ˆ = x   BC  =  CD  =  d    and   CBD B   y   x   d   109 C   D   d

AB  is  perpendicular  to  the  plane.   The  angle  of  elevation  of  A  from  C  is  y.   4.   Answer  the  questions  below:     (a)   Prove  that   AB = 2d cos x tan y                                              (5)     (b)   Given  that   d = 2  units,  x  =  75°  and  y  =  30°,  calculate  the  length  of  AB         without  using  a  calculator.                              (5)   TOTAL  :  20  MARKS   REFERENCES   Name  of  textbook   Author(s)   Publisher   In  this  document  referred  to  as…   1.   Classroom Mathematics Pike,  Kitto,  etc.   Heinemann   Classroom  Maths   2.   Clever Keeping Maths Simple Aird,  Du  Toit,  etc.   Macmillan   Clever  Maths   3.   Maths Handbook and Study Kevin  Smith   Berlut  Books   Maths  Handbook  and  Study  Guide   Guide 4.   Mind Action Series Phillips,  Basson  and  Botha   AllCopy  Publishers   Mind  Action  Series   Mathematics 5.   Platinum Mathematics Bradley,  Campbell  and  McPetrie   Maskew  Miller  Longman   Platinum     6.   Via Afrika Mathematics Botsane,  Smith,  etc.   Via  Afrika  Publishers   Via  Afrika   110